Proof.
The first inequality can be found in [1]}.
We next give the outline of the proof
of the second one.
Note that \(\begin{bmatrix} \!\!\! \otimes ^t A & \otimes ^t B \\ \otimes ^t B^* & \otimes ^t C \end{bmatrix}\) 
is positive semidefinite.
By restricting this block matrix to
the symmetric class of tensor product (see, e.g., [2]}), we know that
\( \begin{bmatrix} \!\!\! \wedge ^t A & \wedge ^t B \\\wedge ^t B^* & \wedge ^t C \end{bmatrix} \) 
